cp's OEIS Frontend

A050157 T(n, k) = S(2n, n, k) for 0<=k<=n and n>=0, where S(p, q, r) is the number of upright paths from (0, 0) to (p, p-q) that do not rise above the line y = x-r.

%I A050157 #12 Dec 21 2017 07:48:14
%S A050157 1,1,2,2,5,6,5,14,19,20,14,42,62,69,70,42,132,207,242,251,252,132,429,
%T A050157 704,858,912,923,924,429,1430,2431,3068,3341,3418,3431,3432,1430,4862,
%U A050157 8502,11050,12310,12750,12854,12869,12870
%N A050157 T(n, k) = S(2n, n, k) for 0<=k<=n and n>=0, where S(p, q, r) is the number of upright paths from (0, 0) to (p, p-q) that do not rise above the line y = x-r.
%C A050157 Let V = (e(1),...,e(n)) consist of q 1's and p-q 0's; let V(h) = (e(1),...,e(h)) and m(h) = (#1's in V(h)) - (#0's in V(h)) for h=1,...,n. Then S(p,q,r) is the number of V having r >= max{m(h)}.
%F A050157 T(n, k) = Sum_{0<=j<=k} t(n, j), array t as in A039599.
%F A050157 T(n, k) = binomial(2*n, n) - binomial(2*n, n+k+1). - _Peter Luschny_, Dec 21 2017
%e A050157 The triangle starts:
%e A050157                                 1
%e A050157                               1, 2
%e A050157                             2, 5, 6
%e A050157                          5, 14, 19, 20
%e A050157                        14, 42, 62, 69, 70
%e A050157                   42, 132, 207, 242, 251, 252
%e A050157                132, 429, 704, 858, 912, 923, 924
%p A050157 A050157 := (n, k) -> binomial(2*n, n) - binomial(2*n, n+k+1):
%p A050157 seq(seq(A050157(n,k), k=0..n), n=0..10); # _Peter Luschny_, Dec 21 2017
%Y A050157 T(n, 0) = A000108(n).
%Y A050157 T(n, 1) = A000108(n+1).
%Y A050157 T(n, n) = A000984(n).
%Y A050157 T(n, n-1) = A030662(n).
%Y A050157 Row sums are A296771.
%Y A050157 Cf. A039599, A050163.
%K A050157 nonn,tabl
%O A050157 0,3
%A A050157 _Clark Kimberling_